{"id":1950,"date":"2021-09-16T17:34:49","date_gmt":"2021-09-16T17:34:49","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1950"},"modified":"2023-12-05T09:27:59","modified_gmt":"2023-12-05T09:27:59","slug":"a-single-population-mean-using-the-normal-distribution-4","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/a-single-population-mean-using-the-normal-distribution-4\/","title":{"raw":"Calculating the Sample Size n","rendered":"Calculating the Sample Size n"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<section>\r\n<ul id=\"list12315\">\r\n \t<li>Calculate the sample size needed for a given error bound for a confidence interval for a population mean based on a normal distribution<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<h2>Calculating the Sample Size <em data-redactor-tag=\"em\">n<\/em><\/h2>\r\nIf researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size.\r\n\r\nThe error bound formula for a population mean when the population standard deviation is known is\u00a0<em>EBM<\/em> = [latex]{\\left ( z_\\frac{a}{2} \\right )}{\\left ( \\dfrac{\\sigma}{\\sqrt n} \\right )}.[\/latex]\r\n\r\nThe formula for sample size is n = [latex]\\dfrac{z^2\\sigma^2}{EBM^2}[\/latex], found by solving the error bound formula for <em data-redactor-tag=\"em\">n<\/em>.\r\n\r\nIn this formula, <em data-redactor-tag=\"em\">z<\/em> is [latex]z_\\frac{a}{2}[\/latex], corresponding to the desired confidence level. A researcher planning a study who wants a specified confidence level and error bound can use this formula to calculate the size of the sample needed for the study.\r\n<div class=\"textbox exercises\">\r\n<h3>Example 7<\/h3>\r\nThe population standard deviation for the age of Foothill College students is 15 years. If we want to be 95% confident that the sample mean age is within two years of the true population mean age of Foothill College students, how many randomly selected Foothill College students must be surveyed?\r\n[reveal-answer q=\"48498\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"48498\"]\r\n<ul>\r\n \t<li>From the problem, we know that <em data-redactor-tag=\"em\">\u03c3<\/em> = 15 and <em data-redactor-tag=\"em\">EBM<\/em> = 2.<\/li>\r\n \t<li><em data-redactor-tag=\"em\">z<\/em> = <em data-redactor-tag=\"em\">z<\/em><sub data-redactor-tag=\"sub\">0.025<\/sub> = 1.96, because the confidence level is 95%.<\/li>\r\n \t<li>n = [latex]\\dfrac{z^2\\sigma^2}{EBM^2}[\/latex] = [latex]\\dfrac{(1.96)^2(15)^2}{2^2}[\/latex] = 216.09 using the sample size equation.<\/li>\r\n \t<li>Use <em data-redactor-tag=\"em\">n<\/em> = 217: Always round the answer UP to the next higher integer to ensure that the sample size is large enough.<\/li>\r\n<\/ul>\r\nTherefore, 217 Foothill College students should be surveyed in order to be 95% confident that we are within two years of the true population mean age of Foothill College students.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it 7<\/h3>\r\nThe population standard deviation for the height of high school basketball players is three inches. If we want to be 95% confident that the sample mean height is within one inch of the true population mean height, how many randomly selected students must be surveyed?\r\n[reveal-answer q=\"734255\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"734255\"]\r\n\r\n35 students\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<section>\n<ul id=\"list12315\">\n<li>Calculate the sample size needed for a given error bound for a confidence interval for a population mean based on a normal distribution<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<h2>Calculating the Sample Size <em data-redactor-tag=\"em\">n<\/em><\/h2>\n<p>If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size.<\/p>\n<p>The error bound formula for a population mean when the population standard deviation is known is\u00a0<em>EBM<\/em> = [latex]{\\left ( z_\\frac{a}{2} \\right )}{\\left ( \\dfrac{\\sigma}{\\sqrt n} \\right )}.[\/latex]<\/p>\n<p>The formula for sample size is n = [latex]\\dfrac{z^2\\sigma^2}{EBM^2}[\/latex], found by solving the error bound formula for <em data-redactor-tag=\"em\">n<\/em>.<\/p>\n<p>In this formula, <em data-redactor-tag=\"em\">z<\/em> is [latex]z_\\frac{a}{2}[\/latex], corresponding to the desired confidence level. A researcher planning a study who wants a specified confidence level and error bound can use this formula to calculate the size of the sample needed for the study.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example 7<\/h3>\n<p>The population standard deviation for the age of Foothill College students is 15 years. If we want to be 95% confident that the sample mean age is within two years of the true population mean age of Foothill College students, how many randomly selected Foothill College students must be surveyed?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q48498\">Show Answer<\/span><\/p>\n<div id=\"q48498\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li>From the problem, we know that <em data-redactor-tag=\"em\">\u03c3<\/em> = 15 and <em data-redactor-tag=\"em\">EBM<\/em> = 2.<\/li>\n<li><em data-redactor-tag=\"em\">z<\/em> = <em data-redactor-tag=\"em\">z<\/em><sub data-redactor-tag=\"sub\">0.025<\/sub> = 1.96, because the confidence level is 95%.<\/li>\n<li>n = [latex]\\dfrac{z^2\\sigma^2}{EBM^2}[\/latex] = [latex]\\dfrac{(1.96)^2(15)^2}{2^2}[\/latex] = 216.09 using the sample size equation.<\/li>\n<li>Use <em data-redactor-tag=\"em\">n<\/em> = 217: Always round the answer UP to the next higher integer to ensure that the sample size is large enough.<\/li>\n<\/ul>\n<p>Therefore, 217 Foothill College students should be surveyed in order to be 95% confident that we are within two years of the true population mean age of Foothill College students.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it 7<\/h3>\n<p>The population standard deviation for the height of high school basketball players is three inches. If we want to be 95% confident that the sample mean height is within one inch of the true population mean height, how many randomly selected students must be surveyed?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q734255\">Show Answer<\/span><\/p>\n<div id=\"q734255\" class=\"hidden-answer\" style=\"display: none\">\n<p>35 students<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1950\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>A Single Population Mean using the Normal Distribution. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/8-1-a-single-population-mean-using-the-normal-distribution\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/8-1-a-single-population-mean-using-the-normal-distribution<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><li>Introductory Statistics. <strong>Authored by<\/strong>: Barbara Illowsky, Susan Dean. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169134,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"A Single Population Mean using the Normal Distribution\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/8-1-a-single-population-mean-using-the-normal-distribution\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"},{\"type\":\"cc\",\"description\":\"Introductory Statistics\",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1950","chapter","type-chapter","status-publish","hentry"],"part":269,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1950","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/users\/169134"}],"version-history":[{"count":4,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1950\/revisions"}],"predecessor-version":[{"id":3763,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1950\/revisions\/3763"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/269"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1950\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=1950"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=1950"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=1950"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=1950"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}